I'm not sure if you will incur additional expenses, as the insurance industry seems to find ways to get the most out of people.
I would phone the insurance company, take notes and names, and ask specifically whst you will be expected to pay.
One other thing that you can try, is to negotiate your obligation with the hospital after the fact.
Often, the hospitals will inflate the charges for an insured patient, and the insurance company's 80% is MORE then they would have received from a cash patient. My insurance was billed $13,000.00 for a standard delivery, that would have cost $4000-$5000, had I been a cash patient. They received more from the insurance company, then I would have been asked to pay if I didn't have insurance at all.
After the insurance payment has been made, ask for an itemized accounting of the charges, and go through them line by line. There WILL be errors, they will be adjusted off of your percentage..or if you are a big enough pain in the ass, they will make an offer to settle.
In my case, a $2500+ co-pay, was negotiated down to $500.
Good luck with the upcoming surgery.